A Set-Oriented Numerical Approach forDynamical Systems with Parameter

Uncertainty

Michael Dellnitz1, Stefan Klus2, and Adrian Ziessler1

1Department of Mathematics, Paderborn University, Germany2Department of Mathematics and Computer Science, Freie Universitat Berlin, Germany

April 26, 2016

Abstract

In this article, we develop a set-oriented numerical methodology which allows toperform uncertainty quantification (UQ) for dynamical systems from a global point ofview. That is, for systems with uncertain parameters we approximate the correspondingglobal attractors and invariant measures in the related stochastic setting. Our methodsdo not rely on generalized polynomial chaos techniques. Rather, we extend classicalset-oriented methods designed for deterministic dynamical systems [DH97, DJ99] tothe UQ-context, and this allows us to analyze the long-term uncertainty propagation.The algorithms have been integrated into the software package GAIO [DFJ01], and weillustrate the use and efficiency of these techniques by a couple of numerical examples.

1 Introduction

The analysis of the influence of uncertainties in complex dynamical systems on the system’sbehavior has gained considerable attention in the last years. In many applications, inputparameters, initial conditions, or boundary conditions are not known exactly and are thusdescribed by probability distributions. The goal is to quantify the effects of these uncer-tainties and their impact on, for instance, stability or performance of the system. Due tothe stretching and folding of the corresponding trajectories, propagating probability densityfunctions through highly nonlinear dynamical systems is particularly challenging [LBR14].

In this article, we consider parameter-dependent discrete dynamical systems assumingthat some parameters are uncertain. The main goal is to develop robust algorithms forthe analysis of the resulting statistical behavior of the system so that we capture the long-term uncertainty propagation. To this end, we compute approximations of the correspond-ing invariant sets and invariant measures using so-called set-oriented numerical methods.These have been developed for the numerical analysis of complex dynamical systems, seee.g. [DH97, DJ99, FD03, FLS10], and used for a host of different application areas such

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as molecular dynamics [SHD01], astrodynamics [DJL+05], and ocean dynamics [FHR+12].Recently, set-oriented methods have been extended to compute attractors for delay differen-tial differential equations [DHZ16]. The basic idea is to cover the objects of interest by outerapproximations created via multilevel subdivision schemes. In this work, we will generalizethese techniques to the context of uncertainty quantification.

A related – though not set-oriented – approach using generalized polynomial chaos (gPC),see e.g. [Sud08, Sul15], has been utilized in [LBR14] for the numerical analysis of uncertaintyin dynamical systems. Long-term uncertainty propagation is accomplished by approximat-ing and composing intermediate short-term flow maps using spectral polynomial bases. AgPC based approach has also been applied in [PS14] in the context of Hamiltonian systemsexhibiting multi-scale dynamics, or in [Xiu07] with an application to differential algebraicequations (DAEs). In contrast to this, our work relies on efficient adaptive subdivisionschemes and transfer-operator based methods. This is the first time a set-oriented approachhas been used for the computation of attractors and invariant measures in this context, andthis allows us to quantify the uncertainty from a global point of view. Technically, the resultsof this article are mainly based on two previous publications: First, we extend the classicalsubdivision scheme in [DH97] to the context of parameter dependence within a compact setΛ. Secondly, our measure computations in Section 3 are based on the theoretical frameworkin [DJ99].

A detailed outline of the paper is as follows: In Section 2, we introduce the notion of(Q,Λ)-attractors. This is an object in state space which contains all the invariant setswithin the compact set Q which can potentially be created by the parameter uncertaintywithin Λ. That is, the particular probability distribution on Λ is not yet relevant forour considerations in this section. Furthermore, we develop an algorithm which allows tocompute outer approximations of (Q,Λ)-attractors (Algorithm 2.5). In Section 3, we assumethat the uncertainty within Λ is given by a certain probability distribution. We modelthis parameter uncertainty with appropriate stochastic transition functions and develop analgorithm for the computation of corresponding invariant measures (Algorithm 3.7). Theuse of small random perturbations [Kif86] allows us to prove a related convergence result,which is essentially an adapted extension of the corresponding result in [DJ99]. In Section 4,we illustrate the use and efficiency of the algorithms by three examples, namely the Henonmap, the van der Pol oscillator and the Arneodo system. Finally, in Section 5, we concludewith a short summary of the main results and possible future work.

2 Computation of (Q,Λ)-Attractors via Subdivision

In this section, we will introduce the notion of (Q,Λ)-attractors, a generalization of theconcept of relative global attractors defined in [DH97]. The goal is to develop a numericaltechnique which allows to identify the region in state space which is potentially influencedby inherent parameter uncertainties.

2

2.1 (Q,Λ)-Attractors

Let us denote by Λ ⊂ Rp a compact subset which represents our set of admissible parametervalues. Then for each λ ∈ Λ we have the dynamical system

xj+1 = f(xj , λ), j = 0, 1, . . . , (1)

where xj ∈ Rn and f : Rn × Λ → Rn is continuous. In order to take the uncertaintyin (1) with respect to λ into account, we consider the corresponding (set-valued) mapFΛ : Rn → P(Rn), where

FΛ(x) = f(x,Λ)

and P(Rn) denotes the power set of Rn.The purpose of this section is to approximate the region in state space covering all the

backward invariant sets which can potentially be generated by λ-distributions with supportin Λ. Therefore, it is not yet necessary to assume that the uncertainty with respect to theparameters is described by a certain distribution. However, we will come back to this inSection 3 where we compute invariant measures for different λ-distributions.

In this section, we develop an algorithm which allows to compute the object

AQ,Λ =⋂j≥0

F jΛ(Q), (2)

where Q ⊂ Rn is a compact subset. We call AQ,Λ the (Q,Λ)-attractor.

Remark 2.1. (a) With (2), we generalize the concept of relative global attractors whichhas first been introduced in [DH97] for dynamical systems of the form

xj+1 = g(xj), j = 0, 1, . . . ,

where g is a homeomorphism. In fact, the global attractor relative to Q is defined by

AQ =⋂j≥0

gj(Q).

(b) In [DSS02, DSH05], relative global attractors have been defined for non-autonomous dy-namical systems. More precisely, given s dynamical systems g1, . . . , gs, one is interestedin computing their common invariant sets. Therefore, the object of interest is

AQ,g1,...,gs =⋂ω∈Ω

⋂j≥1

gωj (Q) ∩Q.

Here, Ω = 1, 2, . . . , sN0 is the space of sequences of s symbols, and for ω = (ωi) ∈ Ωthe map gωj is defined as the composition gωj = gωj−1 · · · gω0.

(c) Our definition in (2) is also strongly related to concepts which have been introduced inconnection with control problems. For instance, the (positive and negative) viabilitykernels as defined in [Szo01] are precisely of this type. However, our analytical setup isdifferent and it is the purpose of this article to utilize the related set-oriented numericalanalysis in the context of uncertainty quantification.

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The following properties of AQ,Λ follow immediately from its definition.

Proposition 2.2. (a) Suppose that the map FΛ : Rn → P(Rn) is one-to-one. Then AQ,Λis backward invariant, i.e.

F−1Λ (AQ,Λ) ⊂ AQ,Λ.

(b) Let B ⊂ Q be a set such that B ⊂ FΛ(B). Then B ⊂ AQ,Λ.

By this proposition, AQ,Λ should be viewed as the set which contains all the backwardinvariant sets of FΛ. Accordingly, in the deterministic context AQ typically consists of allinvariant sets and their unstable manifolds within Q.

2.2 Subdivision Scheme

The numerical developments in this work are based on a modification of the subdivisionscheme introduced in [DH97], where this algorithm is used for an approximation of relativeglobal attractors (see Remark 2.1 (a)). Here, we extend it in order to create coverings of(Q,Λ)-attractors.

This algorithm generates a sequence C0, C1, . . . of finite collections of compact subsets ofRn such that the diameter

diam(C`) = maxC∈C`

diam(C)

converges to zero for ` → ∞. Concretely, given an initial collection C0 with⋃C∈C0 C = Q,

we successively obtain C` from C`−1 for ` = 1, 2, . . . in two steps:

1. Subdivision: Construct a new collection C` such that⋃C∈C`

C =⋃

C∈C`−1

C (3)

anddiam(C`) = θ` diam(C`−1), (4)

where 0 < θmin ≤ θ` ≤ θmax < 1.

2. Selection: Define the new collection C` by

C` =C ∈ C` : ∃C ∈ C` and ∃λ ∈ Λ such that f(·, λ)−1(C) ∩ C 6= ∅

. (5)

We denote by Q` the area in state space covered by C`, that is

Q` =⋃C∈C`

C,

and letQ∞ = lim

`→∞Q`.

Observe that this limit exists due to the fact that the Q` form a sequence of nested compactsets.

4

Remark 2.3. (a) If we replace f(x, λ) by g = g(x) in the subdivision scheme (g a homeo-morphism) then the main theoretical result in [DH97] states that

AQ = Q∞,

see also Remark 2.1 (a).

(b) The result in (a) is also valid in the situation where g is just continuous – and not ahomeomorphism. However, in this case one has to assume additionally that g−1(AQ) ⊂AQ. This result has recently been proved in [DHZ16].

Now we state the main result of this section:

Proposition 2.4. Suppose that the map FΛ : Rn → P(Rn) is one-to-one. Then

AQ,Λ = Q∞.

The proof is in principle identical to the one in [DH97] or [DHZ16], respectively. Thus,we just sketch it here.

Sketch of Proof. The proof consists of the following steps:

(i) AQ,Λ ⊂ Q` for all `. Here, one uses Proposition 2.2 (a).

(ii) Q∞ ⊂ FΛ(Q∞). In this step, the continuity of f is crucial.

Now by (i)AQ,Λ ⊂ Q∞,

and by (ii) and Proposition 2.2 (b)

Q∞ ⊂ AQ,Λ,

yielding the desired result.

Generalizing the approach in [DH97], we use the following numerical realization of thesubdivision scheme for the approximation of the (Q,Λ)-attractor.

Algorithm 2.5. Choose an initial box Q ⊂ Rn, defined by a generalized rectangle of theform

Q(c, r) = y ∈ Rn : |yi − ci| ≤ ri for i = 1, . . . , n,

where c, r ∈ Rn, ri > 0 for i = 1, . . . , n, are the center and the radius, respectively. Dis-cretize Λ = λ1, . . . , λM uniformly and start the subdivision algorithm with a single boxC0 = Q.

1. Realization of the subdivision step: In step (`− 1), we subdivide each box C ∈ C`−1 ofthe current collection by bisection with respect to the s-th coordinate, where s is variedcyclically. Thus, in the new collection C` the number of boxes is increased by a factorof 2 (cf. (3), (4)).

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2. Realization of the selection step: We choose a finite set of test points in each boxCj ∈ C` and replace the condition (5) by

f(x, λk) /∈ Ci for all test points x ∈ Cj and all λk, k = 1, . . . ,M. (6)

A box Ci is discarded if (6) is satisfied for all j.

3. Repeat (1)+(2) until a prescribed size ε of the diameter relative to the initial box Q isreached. That is, we stop when

diam(C`) < εdiam(Q).

Remark 2.6. (a) The distribution of test points in each box C ∈ C` is performed as follows.Observe that each box is defined by a generalized rectangle with a radius r and center cand therefore is the affine image of the standard cube [−1, 1]n scaled by r and translatedby c. Thus, by using this transformation it is sufficient to define the distribution oftest points for the standard cube, e.g. via a uniform grid or a (quasi-)Monte Carlosampling. In our computation, we use the Halton sequence, which is a quasi-randomnumber sequence [Hal64].

(b) Algorithm 2.5 has been developed within the software package GAIO (see [DFJ01]).Thus, as in the deterministic case the numerical complexity depends crucially on thedimension of AQ,Λ. In fact, our experience indicates that we can approximate theseobjects for dimensions up to three even in higher dimensional state space. On the otherhand, it will become very time-consuming to compute AQ,Λ if its dimension is largerthan four.

3 Computation of Invariant Measures on (Q,Λ)-Attractors

We use a transfer operator approach to approximate invariant measures on (Q,Λ)-attractors.This is a classical mathematical tool for the numerical analysis of complicated dynamicalbehavior, e.g. [DJ99, SHD01, FP09, Kol10], and here we use it in the context of uncertaintyquantification.

3.1 Invariant Measures and the Transfer Operator

We briefly introduce the reader to the notion of transfer operator in the stochastic setting. Inthe context of stochastic differential equations, transfer operators and stochastic transitionfunctions have recently been utilized in [FK15]. Also in this paper, we have to stay withinthe stochastic setting in order to take the parameter uncertainty into account. In thefollowing paragraphs, we follow closely the related contents in [DJ99].

As in the previous section, let Q ⊂ Rn be compact and denote by B the Borel-σ algebraon Q.

Definition 3.1. A function p : Q× B → R is a stochastic transition function, if

1. p(x, ·) is a probability measure for every x ∈ Q,

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2. p(·, A) is Lebesgue-measurable for every A ∈ B.

Remark 3.2. Let δy denote the Dirac measure supported on the point y ∈ Q. Thenpλ(x,A) = δf(x,λ)(A), λ ∈ Λ, is a stochastic transition function. This represents the deter-

ministic situation for fixed λ ∈ Λ in this stochastic setup.

We now define the notion of an invariant measure in the stochastic setting. To this end,we denote by M the set of probability measures on B.

Definition 3.3. Let p be a stochastic transition function. If µ ∈M satisfies

µ(A) =

∫p(x,A) dµ(x)

for all A ∈ B, then µ is an invariant measure of p.

The following example illustrates the previous remark that we recover the deterministicsituation in the case where pλ(x, ·) = δf(x,λ).

Example 3.4. Suppose that pλ(x, ·) = δf(x,λ) and let µ be an invariant measure of pλ. Thenwe compute for A ∈ B

µ(A) =

∫pλ(x,A) dµ(x) =

∫δf(x,λ)(A) dµ(x)

=

∫χA(f(x, λ)) dµ(x) = µ(f−1(A, λ)),

where we denote by χA the characteristic function of A. Hence, µ is an invariant measurefor the map f(·, λ) in the classical deterministic sense (cf. [Pol93]).

Definition 3.5. Let p be a stochastic transition function. Then the transfer operator P :MC →MC is defined by

Pµ(A) =

∫p(x,A) dµ(x),

where MC is the space of bounded complex-valued measures on B.

By definition, an invariant measure µ is a fixed point of P , i.e.

Pµ = µ, (7)

and in the remainder of this section we develop a numerical method for the approximationof such measures.

3.2 Stochastic Transition Functions on (Q,Λ)-Attractors

Suppose that the parameter uncertainty on Λ is given by the probability density functionρ : Λ→ R≥0. Then we define the corresponding stochastic transition function qρ as follows:For A ⊂ Rn and each point x ∈ Rn let

Λx(A) = λ ∈ Λ : f(x, λ) ∈ A = f(x, ·)−1(A).

7

The set Λx(A) is measurable for each x ∈ Rn and each A ∈ B since f is continuous, and wedefine the measure of A to be the measure of Λx(A) ⊂ Λ, that is

qρ(x,A) =

∫Λx(A)

ρ(λ) dλ.

Observe that

qρ(x,A) =

∫f(x,·)−1(A)

ρ(λ) dλ =

∫Λ

χA(f(x, λ))ρ(λ) dλ. (8)

By construction, qρ(x, ·) is a probability measure for every x ∈ Rn. Moreover, qρ(·, A) isintegrable by (8) and therefore a stochastic transition function according to Definition 3.1.

In the particular deterministic case where the parameter λ is fixed, we obtain (see Re-mark 3.2)

qρ(x,A) =

∫Λ

χA(f(x, λ))δ(λ) dλ = χA(f(x, λ)) = δf(x,λ)(A) = pλ(x,A).

Here δ(·) is the Dirac delta function.

3.3 Approximation of Invariant Measures

Given a stochastic transition function qρ(x, ·), we now explain how to approximate a corre-sponding invariant measure numerically. For the discretization of the problem, we proceedas in [DHJR97, Kol10] and use characteristic functions χCl

(l = 1, 2, . . . , d) on each box inour covering of the (Q,Λ)-attractor which has been generated by Algorithm 2.5. We denoteby m the Lebesgue measure and define the corresponding probability measures

µCl(A) =

1

m(Cl)

∫AχCl

dm, l = 1, . . . , d.

The transfer operator P is acting on these measures as follows

(PµCl) (A) =

∫qρ(x,A) dµCl

=1

m(Cl)

∫qρ(x,A)χCl

dm =1

m(Cl)

∫Cl

qρ(x,A) dm.

Thus, we can approximate the transfer operator with the following stochastic matrixPd = (pkl) on the box covering C1, . . . , Cd:

pkl =1

m(Cl)

∫Cl

qρ(x,Ck) dm, k, l = 1, . . . , d. (9)

Finally, we approximate the invariant measure corresponding to the stochastic transitionfunction qρ by the stationary distribution of the Markov chain given by Pd. Concretely, weapproximate probability measures ν ∈M by

ν ≈d∑l=1

αl µCl.

8

In order to obtain an approximation of an invariant measure µ, we require(P

d∑l=1

αl µCl

)(Ck) =

d∑l=1

αl µCl(Ck) = αk, k = 1, 2, . . . , d.

Here, we have used the fact that

µCl(Ck) = δkl (δkl the Kronecker-delta)

by the construction of the box collection. That is, for an approximation of an invariantmeasure µ we have to compute the eigenvector αd ∈ Rd≥0 for the eigenvalue 1 of the matrixPd, i.e.

Pd αd = αd

(see also (9)).

Remark 3.6. (a) In the deterministic case, that is qρ(x, ·) = pλ(x, ·) = δf(x,λ), the transi-tion probabilities are given by

pkl =m(f−1(Ck, λ) ∩ Cl

)m(Cl)

.

(b) Numerically, the computation of (9) is realized as follows: For each 1 ≤ l ≤ d, selecttest points x1, . . . , xN ∈ Cl and uncertain parameters λ1, . . . , λM distributed accordingto the probability density function ρ. This yields

pkl =1

m(Cl)

∫Cl

qρ(x,Ck) dm

≈ 1

M ·N

M∑i=1

∣∣ j ∈ 1, . . . , N|f(xj , λi) ∈ Ck∣∣.

(c) The box covering C1, . . . , Cd obtained with the subdivision scheme and the dynamicsinduced by the stochastic transition function qρ yield a directed graph as illustrated inFigure 1. The dynamics on this graph with the transition probabilities in (9) can beviewed as an approximation of the transfer operator P .

We summarize our numerical approach in the following algorithm.

Algorithm 3.7. The strategy for the approximation of an invariant measure correspondingto the stochastic transition function qρ supported on a (Q,Λ)-attractor AQ,Λ can now beformulated as follows:

1. Approximate a (Q,Λ)-attractor AQ,Λ by the subdivision Algorithm 2.5 to obtain a boxcovering C1, . . . , Cd.

2. Use C1, . . . , Cd to compute the discretized transfer operator Pd by (9). If λ is dis-tributed according to the probability density function ρ, we use Remark 3.6 (b) wherethe λ-values are obtained via a (quasi-)Monte Carlo sampling.

3. Compute the eigenvector αd ∈ Rd corresponding to the eigenvalue 1 of Pd to obtainan approximation of an invariant measure µ on AQ,Λ (cf. (7)).

9

8

1 2

3 4

5 6

7

8

12

3 4

56

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Figure 1: Schematic illustration of the graph induced by the transition function qρ on thebox covering. Left: Box covering C1, . . . , Cd and mapping of points from boxCl to box Ck. Right: Resulting directed graph with vertices v1, . . . , vd andedges (vl, vk).

3.4 Convergence Result

We utilize the theoretical framework from [DJ99] to obtain a convergence result for ournumerical approach. Our developments in this section cover the classical deterministicsituation (see Example 3.4). Therefore, we have to consider small random perturbations(cf. [Kif86]) of f(x, λ) so that the transfer operator becomes compact as an operator on L2.That is, in addition to the inherent parameter uncertainty we now introduce a perturbationin state space so that classical convergence theory for compact operators can be applied.

We let B = B0(1) the open ball in Rn of radius one and define for ε > 0

kε(x, y) =1

εnm(B)χB

(1

ε

(y − x

)), x, y ∈ Rn.

We use this function for the definition of a transition density function which allows to takethe parameter uncertainty into account

kε,f (x, y) =

∫Λ

kε(f(x, λ), y)ρ(λ) dλ.

With this we define a stochastic transition function pε in this random context by

pε(x,A) =

∫Akε,f (x, y) dm(y).

Remark 3.8. Observe that∫A

kε(f(x, λ), y) dm(y)→ δf(x,λ)(A) for ε→ 0,

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and therefore

pε(x,A) =

∫Akε,f (x, y) dm(y)→

∫Λ

χA(f(x, λ))ρ(λ) dλ for ε→ 0.

Thus, as expected, we obtain in the limit the stochastic transition function qρ(x,A) in (8).

Due to the small random perturbation, the measure pε(x, ·) is absolutely continuous forε > 0, and the corresponding transfer operator Pε : L1 → L1 can be written as

(Pεg) (y) =

∫kε,f (x, y)g(x) dm(x) for all g ∈ L1. (10)

It is easy to verify that for ε > 0∫∫|kε,f (x, y)|2 dm(x)dm(y) <∞.

Therefore, the transfer operator in (10) as an operator Pε : L2 → L2 is compact.Now let β 6= 0 be an eigenvalue of Pε and let E be a projection onto the corresponding

generalized eigenspace. Then we have the following convergence result which yields anapproximation result for invariant measures in the randomized situation (see Theorem 3.5in [DJ99] and also [Osb75]).

Theorem 3.9. Let βd be an eigenvalue of Pd such that βd → β for d→∞, and let γd be acorresponding eigenvector of unit length. Then there is a vector hd ∈ R(E) and a constantC > 0 such that (βI − P )hd = 0 and

‖hd − γd‖2 ≤ C‖(Pε − Pd)|R(E)‖2.

4 Numerical Results

In this section, we will present numerical results for different dynamical systems. For eachsystem, we assume that there exists one uncertain parameter, denoted by λ. In what follows,let U(Λ) denote a uniform distribution over the set Λ and N (µ, σ2) a Gaussian – if necessarytruncated so that it fits into Λ – with mean µ and standard deviation σ.

4.1 Henon Map

As a first example, let us consider the Henon map [Hen76] defined by

xj+1 = 1− λx2j + yj ,

yj+1 = νxj ,

where λ and ν are parameters. Here, we assume that ν = 0.3 is fixed and λ ∈ Λ = [1.2, 1.4]is an uncertain parameter. The bifurcation diagram in Figure 2 illustrates the dynamicalbehavior for this parameter regime.

We then approximate the (Q,Λ)-attractor AQ,Λ for Q = [−3, 2] × [−0.6, 0.6] using thesubdivision algorithm described in Section 2. In Figure 3, we show corresponding boxcoverings obtained by the algorithm after 6, 10, 14, and 20 subdivision steps.

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Figure 2: Bifurcation diagram for the Henon map with ν = 0.3. Based on an image createdby Jordan Pierce [Wik16].

Given the box covering C1, . . . , Cd obtained by the subdivision scheme, we now useAlgorithm 3.7 for the approximation of invariant measures. The resulting invariant mea-sures for different λ-distributions are shown in Figure 4. Observe that the ”seven-periodicbehavior” in part (b) of the figure is still visible in the result for the symmetrically truncatedGaussian in part (c).

4.2 Van der Pol oscillator

Let us now consider the van der Pol system, given by

x1 = x2,

x2 = λ(1− x21)x2 − x1,

(11)

where λ is the uncertain parameter. Here, we chose the interval Λ = [0.5, 1.5]. For eachλ ∈ Λ the system possesses a stable periodic solution as well as an unstable equilibriumin the origin. In our computation, we approximate the (Q,Λ)-attractor AQ,Λ for Q =[−3, 3]× [−4, 4]. In this example, f(x, λ) is given by the time-T -map φT (x, λ) with T = 4,where φ is the flow of (11). We assume that after this time the full parameter uncertaintyon Λ is again relevant.

Figure 5 (a)–(c) shows box coverings of the reconstruction of the two-dimensional unstablemanifolds which accumulate on the stable periodic orbits at their boundary. Figure 5 (d)shows a box covering of the reconstructed periodic solutions itself. It has been obtainedby removing a small open neighborhood U of the origin, i.e. Q = Q\U , resulting in thedisplayed box covering of A

Q,Λ.

Suppose that λ ∼ N (1, σ2). In the case where σ = 0 (i.e. λ = 1), we only have one stableperiodic solution. In Figure 6 (a), we show the corresponding invariant measure. Figure 6

12

(a) ` = 6 (b) ` = 10

(c) ` = 14 (d) ` = 20

Figure 3: Box coverings of the (Q,Λ)-attractor AQ,Λ of the Henon map obtained by thesubdivision scheme after ` subdivision steps.

(b)&(c) shows the invariant measure for σ = 0.1 and σ = 0.2, Figure 6 (d) the correspondingmeasure assuming that λ is uniformly distributed over Λ.

4.3 Arneodo system

The final example is the Arneodo system [Arn82], which is given by

d3x

dt3+d2x

dt2+ 2

dx

dt− λx+ x2 = 0.

We use the equivalent reformulation as a first-order system

x1 = x2,

x2 = x3,

x3 = −x3 − 2x2 + λx1 − x21.

(12)

13

(a) λ = 1.4 (b) λ = 1.24

(c) λ ∼ N (1.24, 0.0004) (d) λ ∼ U(Λ)

Figure 4: Invariant measure on the (Q,Λ)-attractor AQ,Λ of the Henon map for differentλ-distributions. The density ranges from blue (low density) → green → yellow(high density).

This system possesses the equilibria X1 = (0, 0, 0) and X2(λ) = (λ, 0, 0). The latter isasymptotically stable for λ < 2. At λ = 2, the equilibrium X2(λ) undergoes a supercriticalHopf bifurcation (cf. [KO99]). For λ > 2, points on the two-dimensional unstable manifoldof X2(λ) converge to the corresponding limit cycle on the branch of periodic solutions, wherethe amplitude of the limit cycle grows with increasing values of λ. In Figure 7, we showa bifurcation diagram for the periodic solution for λ ∈ [1.5, 3.4]. For λ ≈ 3.1, the limitcycle loses its stability in a period-doubling bifurcation. We choose Λ = [2.8, 3.4] in orderto quantify the uncertainty in this region where the system undergoes several bifurcations.Analogous to the second example, f(x, λ) is given by the time-T -map φT (x, λ) with T = 2,where φ is the flow of (12). We assume that after this time the full parameter uncertaintyon Λ comes again into play.

Figure 8 (a)–(c) shows successively finer box coverings of the (Q,Λ)-attractor AQ,Λ forQ = [−4, 8]× [−7, 5]× [−7, 5]. In this way, we compute a reconstruction of two-dimensional

14

(a) ` = 10 (b) ` = 14

(c) ` = 18 (d) ` = 18

Figure 5: Box coverings of the (Q,Λ)-attractor AQ,Λ of the van der Pol system obtained bythe subdivision scheme after ` subdivision steps. (a) – (c) Q = [−3, 3]× [−4, 4].(d) Q without a small open neighborhood of the origin.

unstable manifolds of X2(λ) which either accumulate on limit cycles, the period-doubledlimit cycles, or even higher periodic limit cycles, depending on the value of λ. We also obtaina covering of the one-dimensional unstable manifold of X1. For comparison purposes, Figure8 (d) depicts the (Q,Λ)-attractor AQ,Λ for a fixed value of λ = 3.1, i.e. without an underlyingparameter uncertainty. In Figure 9, we finally show two projections of the invariant measureon AQ,Λ, where λ is a Gaussian.

5 Conclusion

In this paper, we introduce the notion of (Q,Λ)-attractors, which can be regarded as ageneralization of relative global attractors for dynamical systems with uncertain parame-ters. These objects capture all the dynamics which may potentially be induced by a given

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(a) λ = 1 (b) λ ∼ N (1, 0.01)

(c) λ ∼ N (1, 0.04) (d) λ ∼ U(Λ).

Figure 6: Invariant measure on the (Q,Λ)-attractor AQ,Λ of the van der Pol system fordifferent λ-distributions. The density ranges from blue (low density) → green→ yellow (high density).

parameter distribution on Λ. We then use a classical transfer-operator approach to computeinvariant measures on (Q,Λ)-attractors which are related to different λ-distributions. Thetechnical framework from [DJ99] allows us to obtain a related convergence result in thecontext of small random perturbations. The numerical examples illustrate the fact that thetechniques are very well applicable to low dimensional dynamical systems with parameteruncertainty.

So far, we considered only dynamical systems with one uncertain parameter, i.e. Λ ⊂ R.Future work will include analyzing systems with multiple uncertain parameters and also thescalability of the proposed algorithms. Due to the curse of dimensionality, analyzing high-dimensional systems or systems with a large number of uncertain parameters is in generalchallenging. Finally a set oriented numerical method adapted to the analysis of uncertaintyquantification with respect to initial conditions will also be developed.

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Figure 7: Schematic bifurcation diagram for the Arneodo system for λ ∈ [1.5, 3.4] (Hopfbifurcation and beginning of period doubling sequence).

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